is found to be troublesome, due to the complexity of spacetime metric. However the calculation turns out to be much simpler if we consider the complex path method in \((1+1)\)  dimension [1]}, where the case of Taub-NUT black hole is worked out in [2]} and for a Vaidya black hole in [3]}. Note that the calculation in [2]} applies to a general stationary and axial symmetric black hole spacetime metric
\(ds^2 = - \tilde{f}\left( {r,x} \right)dt^2 + \frac{{dr^2 }}{{\tilde{g}\left( {r,x} \right)}} + \tilde{C}\left( {r,x} \right)h_{ij} \left( {r,x} \right)d\tilde{x}^i d\tilde{x}^j \,,\) 
